REU Meeting - 2026-07-27

This following is a brief summary of our research meeting on 2026-07-27.

Meeting summary

We discovered various applications of Lie theory, including in particle physics, relativity, quantum computing, machine learning and robotics. Woo!

Tasks for next meeting

It's time to connect the categories of Lie groups and Lie algebras.

• Describe a functor $\mathbf{L}\mathbf{i}\mathbf{e}\mathbf{G}\mathbf{r}\mathbf{p}\to \mathbf{L}\mathbf{i}\mathbf{e}\mathbf{A}\mathbf{l}\mathbf{g}$. Make sure to describe both the object function and the arrow function.
• Does this functor have any nice properties; e.g., does it preserve products?
• What does Lie's Third Theorem say about this functor?
• What is the adjoint representation of a group, $G$?
• If $G\to \mathrm{Gl}(\mathfrak{g})$ is the adjoint representation of $G$, how do we get a Lie algebra morphism $\mathfrak{g}\to \mathfrak{gl}\mathfrak{(}\mathfrak{g}\mathfrak{)}$?

References

nLab